Curie temperature of Kondo lattice films with finite itinerant charge carrier density J. Kienert and W. Nolting Festk¨orpertheorie, Institut fu¨r Physik, Humboldt-Universit¨at zu Berlin, Newtonstr. 15, 12489 Berlin, Germany WepresentamodelstudyofferromagneticfilmsconsistingoffreeBlochelectronscoupledtolocal- ized moments (Kondolattice films). By mapping thelocal interaction onto an effective Heisenberg Hamiltonianweobtaintemperatureandcarrierdensitydependentexchangeintegralsmediatingthe 7 interaction between local moments via the conduction electrons. The non-perturbative approach 0 recovers analytically the weak-coupling RKKY interaction and yields convincing numerical results 0 in the strong coupling (double exchange) regime. The Curie temperature is calculated for various 2 coupling strengths, band fillings, and numbers of layers. The results are compared with total en- n ergy calculations. Wediscuss theinfluenceof charge transfer between film layers and of anisotropy a on the Curie temperature. The model we investigate is considered relevant for the understanding J of the basic magnetic properties of manganites, diluted magnetic semiconductors, and rare earth 6 substances as, e.g., Gadolinium. 1 ] I. INTRODUCTION diagram, the most prominent phenomenon of which is l e colossalmagnetoresistance.3,4Althoughforamorequan- r- The Kondo lattice model (KLM) is one of the proto- titativedescriptionofmanganitematerialsadditionalin- t type models in solidstate physics wheneverthe coupling gredients like orbital physics, electron-polaron interac- s . of itinerant charge carrier spins with immobile localized tion,anda strongHubbard-likeCoulombinteractionare t a magnetic moments has to be considered. There are vari- needed,numerousresearchworkonthe subjecthasbeen m ousquitedifferentclassesofmaterialsforwhichtheKLM carried out using the simple one-orbital model (1).3 - isusedtodescribetheelectronicandmagneticproperties More recently attention was directed to the diluted d at least in principle. magnetic semiconductors (DMS) that are thought to n In its simplest single-band version the KLM Hamilto- have the potential for promising technical applications o c nian reads in microelectronics.5,6 Substituting transition metal im- [ purities like Mn in a semiconducting host (often III/V 1 H = (Tij −µδij)c†iσcjσ −J Si·σi . (1) like GaAs) localized magnetic moments are introduced ijσ i with the consequence of ferromagnetic ordering of the v X X randomly distributed maganese cation spins that inter- 9 8 The first term describes non-interacting electrons (c(iσ†) act via their (ferromagnetic, intermediate) coupling to 3 annihilates(creates)anelectronofspinσ =↑,↓atlattice valence and impurity band holes. 1 site i) with hopping integralsT . σ is the electronspin Anotherimportantfieldofapplicationofthe FKLMis ij i 0 andS thelocalizedspinoperator,andbothareexchange the theoretical description of rare earth substances like i 7 coupled by J. EuandGdandtheircompounds. Hereitinitiallyserved 0 DependingonthesignofJ oneisconfrontedwithquite to explain the famous red-shift of the absorption edge / t differentphysics. Numerousstudies havebeen carriedto of the optical 4f − 5d transition in the ferromagnetic a m understand the physics of heavy fermion systems. Here semiconductorEuO.7 CombinedLDA/many-bodycalcu- the couplingbetweenelectronsandlocalizedspinsfavors lationsbasedontheFKLMyieldedrealisticvaluesofthe - d an antiparallel configuration (J < 0) leading to a com- CurietemperatureforGd8andpredictedsurfacestatesin n petition between RKKY interaction and spin screening EuO-films9. Doping Gd into EuO makes for an attrac- o effects. tive means to tune the charge carrier density and thus c Inthisworkwewanttofocusonafavoredparallelspin the carrier-inducedcoupling among the magnetic ions.10 : v alignment (J > 0, ferromagnetic Kondo lattice model, Havingthesamebenefit,Gd-dopedGaNwasreportedto Xi FKLM). The FKLM was introduced to model the mag- yield high TC ferromagnetism above 300 K.11 netism in some manganese compounds (”manganites”). In all of the above-mentioned substances, especially r a In the framework of a simple two-site model this kind of regarding possible technical applications in the future, interactionwasfound to resultfrom the so-calleddouble it is important to understand and make use of the spe- exchange lending its name to the lattice version in the cial physical properties due to reduced dimensionality. strong coupling regime.1,2 Several recent works focussed on the role of magnetic In the manganites the 5 Mn d-shells are split by the anisotropyeffectsinthinfilmsofmanganites12orofDMS crystalfieldintothreedegeneratet -orbitalsforminglo- films like (Ga,Mn)As13. Gd films are investigateddue to 2g calized spins of S = 3 which interact via Hund’s rule theirpotentialemploymentinspintronicsaswellasmoti- 2 with itinerant electrons stemming from the remaining vatedbythe”evergreen”issueofapossiblyenhancedsur- twodegeneratee -orbitals. Due to the Jahn-Tellereffect face Curie temperature.14,15,16 It is furthermore a well- g and the strong Coulomb interaction among the conduc- knownfactthatthe Curietemperature stronglydepends tion electrons the manganites exhibit a complex phase on the film thickness. Due to the broad relevance of the 2 FKLM in current research on magnetism and electronic II. ELECTRONIC SELF-ENERGY correlationsanddue totheimportanceoffilmstructures it is worthwhile investigating the dependence of the fer- In second-quantized notation and introducing Greek romagnetic transition temperature on the model param- indicestonumberthelayerstheHamiltonian(1)becomes eters. H = H +H (2) 0 J = (Tαβ−µδαβ)c† c (3) ij ij iασ jβσ ijσ Xαβ From the theoretical point of view our treatment of 1 FKLM films that is divided up into two parts. First we − 2J zσSizαniασ +Si−ασc†iασciα−σ . (4) solvetheproblemforthe electronicself-energywithinan Xiασ(cid:16) (cid:17) equation of motion approach that is based on a decou- S+,− =Sx ±iSy , ↑(↓)=+ (−), z =1, z =−1, and pling scheme fulfilling non-trivial limiting cases. In the iα iα iα ↑ ↓ Sx,y,z are the cartesian components of the localized spin second step we map the interaction onto an effective lo- iα operator S . n is the particle density operator. calizedspinHamiltonianbyintegratingouttheelectronic iα iασ Inthefollowingweapplyanequationofmotion(EOM) degreesof freedom. This results in effective exchangein- approach based on a decoupling scheme already applied tegralswhich incorporatethe interactionof the localized tosemiconductingKondolatticefilms(EuO9andEuS19) moments via the conduction electrons beyond conven- and to bulk Gd8 in combined LDA and many-body tional RKKY theory. They depend on temperature and calculations (moment conserving decoupling approxima- carrier density. In sum we end up with a self-consistent tion, MCDA). For details of the method we refer the theorywhichdoesnotrequiretheassumptionofclassical readertocorrespondingmodelstudies.20,21,22 Afterwrit- core spins and which is employable at all temperatures. ing down the EOM of the one-particle Green function Gαβ(E)=hhc ;c† ii and the generatedhigher Green ijσ iασ jβσ functions a decoupling is performed that is guided by some non-trivial cases, as, e.g., the ferromagnetic semi- Results onthe magnetizationandcriticaltemperature conductoratT=0. Inthek-spacedefinedwithinthefilm of ferromagnetic KLM model films with finite band oc- plane and using matrix notation one can write: cupation have already been reported in Ref. 17. How- ever, in the present work we make use of a more sophis- Gkσ(E)=[EI−ǫ(k)−Σkσ(E)]−1 . (5) ticated approach for the fermionic subsystem and carry ǫ(k)isthe hoppingmatrix. Alocalapproximationofthe out a wider and considerably more detailed investiga- self-energy Σαβ(E) →δ δ Σ (E) is performed. This tion of the Curie temperature. A similar system is also ijσ ij αβ ασ corresponds to neglecting magnon energies in the three- treated in Ref. 18 based on a method guided by the dimensional case23 and is justified by the fact that these same spirit. However, the theory of the electronic sub- energiesareusuallyordersofmagnitudesmallerthanthe systemthereisonamean-fieldlevelonlyandjustnearest exchange coupling J and the electron bandwidth W. neighborHeisenberg interactionis considered,in spite of From (5) one immediately obtains the one-particle lo- the underlying RKKY-like exchange being long ranged. cal density of states (LDOS): What’s more the effective exchange integrals derived in Ref. 18intermsofthesusceptibilityoffreeelectronsare 1 proportional to J2 and thus fail to give the appropriate ρασ(E)=−π¯hImGαiiασ(E+i0+−µ). (6) strong-couplingbehaviorwheretheCurietemperatureis It shouldbe mentionedthat in generalthe MCDA ap- independent of J. As far as the numerical evaluation is proach does not capture Kondo scaling and especially concernedthethicknessdependenceoftheCurietemper- violates the Luttinger theorem ImΣ (E = ǫ ) → 0.24 ature is analysed. σ F Although this might well be due to our approximative scheme questions if and in which parameter regimes the ferromagnetic Kondo lattice model is a Fermi liquid still remain open. Nevertheless we would consider it worth- This paper is organizedas follows. In the next section while to investigate the low temperature Kondo physics we briefly summarize our method to solve the KLM for withinthe frameworkofourtheory. Inthepresentwork, the electronic self-energy. Then we derive an effective however, we rather want to focus on the magnetic tran- Heisenberg Hamiltonian which results from integrating sition temperature. out the fermionic degrees of freedom. In the numeri- The self-energy matrix Σ (E) depends on various ex- σ calresultssectionwediscuss the temperature dependent pectationvaluesofpurefermionic,mixedfermionic-spin, quasiparticleexcitationspectrumandchargetransferbe- and pure localized spin character: forewepresentanextensiveanalysisoftheCurietemper- atureanditsdependenceonthevariousmodelparamters Σασ =F(hnασi,hSα−σc†ασcα−σi, hSαznασi, (7) like, e.g., the magnetic anisotropy strength. hSzi, h(Sz)2i, h(Sz)3i, hS+S−i). α α α α α 3 The first two types can be calculated self-consistently conventional RKKY interaction is reproduced. In order within the MCDA using the corresponding Green func- to include spin scattering effectively we dress the free tions and the spectral theorem25. The localized spin propagator in (12) and replace it by the full propagator correlation functions like, to lowest order, the layer- G (E). Within this modified RKKY approxima- k+qαβσ dependent magnetization hSzi need further considera- tion (MRKKY) the exchange integrals in the effective α tion. In order to evaluate these we map the interaction Hamiltonian term(4)onto aneffective couplingbetweenthe localized spins, which is carried out in the next section. Heff =− JˆαβS S (13) J ij iα jβ ijαβ X III. EFFECTIVE LOCALIZED SPIN are given by HAMILTONIAN +∞ J2 It is well known that using perturbation theory (4) Jˆαβ(q) = Im dEf(E)G(0)αβ(E)Gαβ (E) 4πN kσ k+qσ leadstotheso-calledRKKYinteraction.26,27Thissecond Xkσ −Z∞ order interaction between localized spins is long ranged (14) and yields an oscillatory behavior of the exchange cou- pling which is mediated by uncorrelated electrons. It where f is the Fermi function and the sum is over the appears highly desirable to have such a mapping with- first Brillouin zone. out being restricted to the weak coupling regime. This Within the Tyablikov approximation27,28 the imagi- requires finding an effective Hamiltonian nary part of the transversal layer-diagonal spin Green Heff = hH i(c) (8) function (spectral density) can be written as J J = −2JN iXαkσqσ′ e−iqRαi (Siα·σ)σσ′hc†k+qασckασ′i(c), − π1ImhhSk+α;S(−−k)αii=2hSαziXγ ηkαγδ(E−Eγ(k))(15) where N is the number of lattice sites in one layer and where the spectral weights ηαγ and energy poles E (k) k γ the superscriptc formally indicates that the averagingis have to be evaluated numerically. For a monolayer we performed in the fermionic subspace only. The method can drop all Greek indices, η = 1, and we get the well- k wepresentinthissectionhasbeenappliedbeforetobulk known spin wave energies Kondolattice systems.8,22 We generalizeit intoa matrix formulation which allows us to consider film structures, E(k)=2hS i(Jˆ(0)−Jˆ(k)). (16) z but is also relevant whenever sublattice decompositions are necessary, for instance when dealing with antiferro- Applying the Callen method27,29 to superlattices the magnetic configurations. layer-dependent magnetization, In order to obtain the expectation value in (8) we in- troduce the modified Green function (1+ϕ +S)ϕ2S+1+(S−ϕ )(1+ϕ )2S+1 hSzi= α α α α (17) Gˆσk′,kσ+qαβ(E)=hhckασ′;c†k+qβσii(c) . (9) α (1+ϕα)2S+1−ϕ2αS+1 The EOM for Gˆ reads and other higher order spin correlation functions can be obtained using the Bose-like distribution function (Eδγα−ǫαkγ)Gˆσk′,kσ+qγβ(E)=δq0δσσ′δαβ (10) 1 ηαγ −2JNXγ ei(k−p)Rαi (Siα·σ)σ′σ′′Gˆσp′,′kσ+qαβ(E). (11) ϕα = N Xk,γ eβEγ(kk)−1 . (18) ipσ′′ X (5),(14), and (17) represent a self-consistent system of In principle, this equation can be solved iteratively. Ap- equations that can be solved for the one-particle Green plying the spectral theorem then yields the expectation function matrix Gkσ(E) and the magnetization hSαzi. value in (8). This correlation function is an operator Beforeproceedingtothenumericalevaluationwehave since weareworkinginthe fermionicsubspace,resulting to reconsiderour effective Hamiltonian(13). It is known in an effective interaction among the localized spins S that for low-dimensional systems anisotropies can be- i only. come very important and even a necessary condition for We haveto findamanageableapproximationfor Gˆ on magnetic ordering at finite temperature.30,31 We there- the right hand side of (10). It can easily be shown that fore include a single-ion anisotropy term in the non-interacting limit, H =−Kα (Sz )2 . (19) Gˆpσ′,′kσ+qαβ(E)→δσ′′σδp,k+qG(k0+)qαβ(E), (12) A 2 iα iα X 4 The physical background of this magnetic anisotropy is 2 T=0 ↑ surface spin-orbit coupling, which usually is some order of mag- center nitudes smaller than the exchange coupling between lo- 0 calized spins. A positive K2α favors an out-of-plane easy ↓ axis, i.e. perpendicular to the film plane. -2 We treat this term in the Anderson-Callen 2 T=0.5T n=0.2, J=0.2 eV, T =174 K approximation32 and decouple the higher Green C C σ function generated by (19) in the following manner: α 0 ρ hh S+,H ;S−ii = hhS+Sz +Sz S+;S−ii -2 iα A − jβ iα iα iα iα jβ (cid:2) (cid:3) ≈ ΦαhhSi+α;Sj−βii, (20) 2 T=TC 0 Φ =2hSzi 1− 1 S(S+1)−h(Sz)2i . (21) -2 α α 2S2 α (cid:18) (cid:0) (cid:1)(cid:19) -0.5 0 E-µ0 .[5eV] 1 1.5 We can drop the site index i due to lateral translational invariance. The single-ionanisotropyacts as aneffective FIG.1: One-particledensityof states ofa5-layerfilm at dif- fieldΦ coupledtoSz. Notethatthisdecouplingisvalid α α ferent temperatures. Parameters: n=0.2, J =0.2 eV, S = only if the magnetization is parallel to the z-direction. 7/2. CriticaltemperatureTC =174K.Chargeneutralcalcu- In order to ensure this a rotation of the coordinate sys- lation. temmightberequiredbeforethedecoupling,forinstance when an arbitrary oriented magnetic field is present.33 Forbulkoramonolayer(α=1)itispossibletoderive A. Electronic properties asimpleformulafortheCurietemperatureT . Expand- C ing (17) and (18) in the vicinity of TC one easily finds 1. One-particle excitation spectrum −1 3k T 1 We start with the discussion of the local one-particle B C = (22) S(S+1)N Xq K2γ+2(Jˆ(0)−Jˆ(q)) T=TC dteemnspiteyraotufrsetsatbeest.weFeing.T1=sh0owasndthTeCL.DTOhSeaCtudriieffetreemnt-  (cid:16) (cid:17) perature has been self-consistently computed. where The general picture for a layered system is the same as for the bulk Kondo lattice model.22 At low tempera- Φ(T) 2(2S−1) γ = lim = (23) turethespin-upspectrumbasicallyconsistsoftherigidly T→TC hSzi 3S shifted non-interacting electron energy band. At T = 0 and in the insulating limit (band occupation n=0) this is a constant and depends on the specific decoupling of is a rigorous result.25 For the spin-down LDOS the sit- the anisotropy Green function (20).34 uation is different. Besides a band at higher energies For α>1 the spin wave energies are no longer known corresponding to an antiparallel coupling of conduction inanalyticalformasin(16). Onecanthen,however,use electrons and localized spins there is a scattering part in (17) to determine the critical temperature. the energy range of the ↑-band reflecting the fact that a ↓-electron can flip its spin by creating a magnon in the localized spin subsystem. With increasing temperature IV. NUMERICAL RESULTS spin symmetry in the spectrum is gradually established. Thisisaccompaniedbyareductionofthebandwidthdue All numerical results have been obtained for simple to reduced effective hopping by spin scattering. For in- cubic (sc) (100) films. The nearest-neighbor hopping termediate coupling strengths J ≈ W as in Fig. 1 this integral t in the tight binding approximation was cho- leads to a temperature-induced opening of a gap in the sen according to a bulk bandwidth W = 1 eV, i.e. excitationspectrum. From the zerobandwidth limit one t = −0.083 eV. The value of the exchange coupling is can learn that the distance between the two correlated assumed to be homogeneous and isotropic for all sites bands roughly scales as ∼J(S+ 1). 2 and layers of the films, as already implied by (4). Fur- Apart from these facts already known from bulk re- thermore for the anisotropy constant we set Kα = K sultsonealsoobservestypicalfeaturesofreduceddimen- 2 2 for all layers. The magnitude of the localized spin is sionality. The number of van-Hove singularities which S = 7/2. To keep notation simple we write n for aremostpronouncedatlowtemperaturesisindicativeof (α,σ) the respective expectation values hn i. n denotes the the number of layers and therefore for the finiteness of (α,σ) average electron density. the film system. A second important observation is the 5 0.04 0.78 J=0 0.02 J=0.1 eV J=0.5 eV α=1 0.76 n - α=1 0 nα α=3 n 0.74 α=2 -0.02 -0.04 0.72 0 0.2 0.4 n 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 J/W FIG. 2: Charge transfer in a 5-layer film at T =0, hSzi=S FIG. 3: Charge transfer in a 5-layer film at T =0, hSzi=0. (ferromagneticsaturation). Whereasthecarrierdensityinthe The horizontal line at n = 0.75 indicates the total average surface layer is always reduced in thenon-interacting system occupation number. The choice of the energy scale refers to there is an occupation above average for large enough n in thefree simple cubicbulk bandwidth W =12t. thecorrelated system. reduced effective bandwidth of the surface compared to a reduced or enhanced occupation in the surface layer is the center layer LDOS. This can be directly traced back linkedtothechemicalpotentialbeinginthelowerorup- to the lower variance of the surface LDOS: in the non- per halfofthe LDOS,respectively. As there is a(mainly interacting limit ∆2ρ = qt and q /q = 5/6 for ↓-) transfer of spectral weight to higher energies for fi- 0 surface bulk a sc(100)geometry. The effect is thus not correlationin- nite J the chemical potential crosses the maximum of duced and more pronounced for ”open” film geometries. the lower band at values n < 1. One finds the quali- ThebandedgesofthesurfaceandcenterLDOS,however, tatively same and quantitatively similar behavior in the are the same. spin disordered phase where hSzαi=0. A second point is the dependence of the charge trans- fer on the coupling strength J. The interaction term (4) 2. Charge transfer alone energetically favors single occupation of a lattice site for n <1. This can be easily seen in the zero band- width limit.25,35 It can thus be expected that a finite J The broken translational symmetry is furthermore re- suppresseschargetransfer. ThisisindeedthecaseasFig. sponsible for the occurence of charge transfer, i.e. the deviation δαn of the layer dependent band filling from 3showswheretheelectrondensityforthedifferentlayers of a 5-layer film is plotted as a function of the exchange the average occupation number: coupling. The change of sign of δαn is again caused by 1 thesamemechanismaspointedoutabove. Givenaband δαn=n −n= (n − n ). (24) α ασ γσ occupation of n < 1 the chemical potential moves from N L Xσ Xγ the lower half of the Bloch bands (J = 0) to the upper half of the lower quasiparticle subband (J >0). N is the number of layers. Charge transfer is already L We observe that even J → ∞ does not lead to com- present in the free electron film system and is due to plete charge neutrality and that at large J the higher the smaller effective bandwidth of the surface layer (oc- occupation of the surface layer goes with an almost ho- cupation n ). From this follows directly that in order to s mogeneous charge distribution in the inner layers of the ensurethermodynamicequilibrium,i.e. acommonchem- film. ItisalsointerestingtonotethatatJ ≈0.1thereis ical potential µ, one has n <n below and n >n above s s a quasi-homogenous charge distribution which coincides half-filling (n=1). with the change of sign of δαn. Switching on the coupling of the electron spins to the localized spins the question arises how this behav- One can summarize that the effect of charge transfer ior changes. Fig. 2 shows the difference of the surface is highest at low couplings J and low average band oc- andthe averageoccupationnumberas afunction ofn at cupations n, where it can amount to almost 30% (Fig. T = 0 and ferromagnetic saturation. Whereas the elec- 2 at n = 0.1). It appears to be of minor importance in trondensityinthesurfacelayern isalwaysbelowthe the strong coupling regime, especially at higher charge α=1 averagedensityninthenon-interactinglimit(J =0)one carrier densities. also finds δαn > 0 at n < 1 for J > 0. The explanation All the findings we have discussed in this section are follows the same reasoning as above for the free system: qualitatively the same for other film thicknesses. As al- 6 400 0.02 α = 1 n=0.01 α = 2 α = 5 charge neutral 0.01 10 α = 5 charge transfer 300 α = 10 0 bulk ∆U ] 00 0.02 ]-0.01 ∆Ukin n = 0.2 K V ∆U [C200 [e pot T U ∆ 0.01 100 0 n = 0.2 -0.01 n = 0.6 0 -0.02 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 J/W J/W FIG.4: Curietemperatureforvariousnumbersoflayers. Pa- FIG.6: Total,kinetic,andpotentialenergydifferences∆U = rameters: n=0.2,K2 =10−6 eV. There is no significant in- UPM(T=0)−UFM(T=0) between the paramagnetic and the fluence of charge transfer on the critical temperature. Inset: ferromagnetic state as a function of the exchange coupling Curietemperatureofamonolayerforalowbandoccupation. J for a monolayer. The negativevalues of ∆U at small J for n=0.6 indicate that ferromagnetism is unstable. ready mentioned charge transfer effects are more pro- B. Curie temperature nounced in ”open” surface geometries with a stronger relative reduction in the number of nearest neighbors of 1. Whole J-range a surface atom. However, the order of magnitude of the effect is the same. Our non-perturbative theory allows for an evaluation Of course in real systems, Coulomb interaction pre- ofKLMfilmsatallcouplingstrengthsJ. Inwhatfollows vents charge separation on a macroscopic scale. For the anisotropy constant is much smaller than the other the present purpose it is sufficient to discuss how charge energyscalesinourmodel,K <<W,J. Theinfluenceof transferaffects the ferromagnetictransitiontemperature 2 theanisotropyontheCurietemperaturewillbediscussed of Kondo lattice films. To this end we will compare in the last subsection in more detail. results based on calculations with and without charge Figs. 4 and 5 display the ferromagnetic critical tem- transfer. Enforcing charge neutrality means we evalu- perature as a function of the exchange coupling J for ated the layer-dependent centers of gravity of the Bloch various film thicknesses. The inset in Fig. 4 shows T bandsself-consistentlysuchthatn =nforallα. Where C α of a monolayer for a small band occupation n = 0.01. not otherwise stated the following results were obtained Here one recognizes the typical T ∼J2-behavior of the for charge neutral KLM films. C perturbational RKKY interaction for small J. Increas- ing the electron density induces a critical interaction J c below which ferromagnetism is not stable. For a given 400 number of layers J increases with n. It has been shown c α = 1 before analytically that there is a minimum critical in- α = 2 α = 5 teraction strength for stable FM in the KLM in infinite 300 α = 10 dimensions.37 bulk To discuss this point further we consider the internal K] energy per site U: [ 200 C T U =hHi=hH0i+hHJi=Ukin+Upot . (25) It can easily be calculated with the LDOS (6): 100 n = 0.6 1 +∞ U = dEf(E)Eρ (E). (26) γσ N 0 L Xγσ Z−∞ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 J/W We start from F = U − TS where F is the free en- ergy and S is the entropy. By comparing the differ- FIG.5: Curietemperatureforvariousnumbersoflayers. Pa- ence of the internal energy at T=0 between the ferro- rameters: n=0.6,K2 =10−6 eV. magnetic (FM) and paramagnetic (PM) state, ∆U = 7 0.04 0.03 0.03 3T [K] 0C0 α = 1250 k T MRKKY 0.025 ∆∆∆UUUkin kT [eV]B0.02 00 0.2 0.4 1 0.6 0.8 1 ∆∆∆BUUUkpCiont U [eV]00.0.0125 pot , 0.01 ∆ U 0.01 ∆ 0 0.005 -0.010 0.2 0.4 0.6 0.8 1 00.01 0.015 0.02 0.025 0.03 n k T [eV] B C FIG. 7: Total, kinetic, and potential energy differences be- FIG. 8: Total, kinetic, and potential energy differences be- tween the paramagnetic and the ferromagnetic state ∆U = tween the paramagnetic and the ferromagnetic state ∆U = UpaPtMio(nT=foTrCa)m−oUnFoMla(yTer=i0n)tahseastfruonncgticoonupolfintgherebgaimnde.ocTchue- UPM(T=TC)− UFM(T=0) as a function of the Curie tem- perature in the strong coupling regime. The film thickness Curie temperatureobtained with theMRKKYtheory is also is an implicit parameter and increases from left (monolayer) included for comparison. Inset: MRKKY Curie temperature f1o0r−v6aerVio.us numbers of layers. Parameters: J = 1 eV,K2 = t1o0−r6igehVt.(NL = 15). Parameters: n = 0.5,J = 1 eV,K2 = fer on T compared to the charge neutral case as Fig. 4 C UPM(T=0)−UFM(T=0),wecanevaluatethestabilityof demonstrates for a 5-layer film.38 the ferromagnetic state against paramagnetism. In Fig. 6 the kinetic, potential, and total energy differences are shown for a monolayer at T=0. The magnetic stability 2. Double exchange regime forlowtomoderateJ isgovernedbyacomplexinterplay between U and U . Whereas at n=0.2 it is the po- kin pot In the strong coupling (double exchange) regime tentialenergywhichfavorsthe(ferro)magneticstateitis (JS >> W) one observes that T runs into saturation C bothenergieswhichmakethe ferromagneticstate unsta- for large enough J at all electron densities and numbers ble forhighern, leadingtoa criticalinteractionstrength of layers. Once more we will contrast the results ob- asintheself-consistentcalculationbasedontheeffective tained by the MRRKY method with energy considera- Heisenberg model before. The fact that in terms of the tions. Whereas before we were interestedin the stability groundstateenergythereis noJ atn=0.2contraryto c of ferromagnetism at T=0 we want to be more quanti- the results in Fig. 4 suggests that other magnetic corre- tative in the following comparisonof the Curie tempera- lations such as non-commensurate or antiferromagnetic tures. Henceweevaluatetheinternalenergyofthepara- becomeimportant,reducingfurthertheparameterrange magnetic state at T as the energy scale for the critical C of FM stability. temperature is set by the change in total energy when Furthermore it is remarkable that the critical interac- going from the ferromagnetic to the paramagnetic state tion strength is smallest for a monolayer whereas films T ∼ ∆U = ∆U(T ) = U (T=T )−U (T=0). In C C PM C FM with α > 1 behave rather bulk-like as far as Jc is con- the double exchange regime TC is essentially determined cerned. Taking into consideration the energy scales of bythekineticenergy: inthelocalframetheelectronspin the KLM one would expect this tendency. Quite gener- is oriented parallel to the localized spin and the poten- ally the physics should be governed by the ratio J/W. tial energydoes not change muchbetween the ferromag- Hence a smaller bandwidth due to a reduced number of neticandtheparamagneticstate;forJ →∞itstaysthe layersimplies asmallerJ aroundwhichthe intermediate same. On the other hand hopping of electrons strongly coupling regime is located. In addition the reduced bulk depends on the magnetic configuration and is somehow FMregionmightindicatefavoredantiferromagneticcon- blocked when the localized spins are disordered. For the figurationswhicharenotexistentinthe2Dcaseformere bulk KLM the relationship between T and ∆U has C kin geometrical reasons. already been analysed.37,39 For comparison we performed calculations where The inset of Fig. 7 shows the Curie temperature as charge transfer was permitted. Allowing charge transfer a function of the band occupation n for strong cou- means that we used equal center of gravities of the non- pling. We obtain ferromagnetism for a wide range of interacting local density of states Tα =T for all layers. n up close to half-filling and a symmetry approximately 0 0 Wedidnotfindanynoteworthyinfluenceofchargetrans- aroundquarter-fillingforallfilms. T increaseswith the C 8 300 J = 1.0 eV n=0.2 J = 0.5 eV 250 J = 0.2 eV 200 J = 0.16 eV J = 0.14 eV 200 J = 0.1 eV K] K] [C150 [C T T 100 0.04 100 K] 0.03 T [1/C0.02 1/ 0.01 50 0 0 5 10 15 -lg(K/eV) 2 0 0 0 1 2 3 4 5 6 7 8 9 10 11 0 0.2 0.4 0.6 0.8 1 Number of layers J/W FIG. 10: Curie temperature of a monolayer at different FIG.9: Curietemperatureasafunctionofthefilmthickness. anisotropy strengths for n=0.6 (from bottom to top: K = Lines are guides to the eye. Parameters: n = 0.2,K = 2 2 10−6 eV. 1, 10, 100 µeV). Varying the anisotropy has a considerable effect on TC in the strong coupling regime but only slightly changes Jc. Inset: Curie temperature as a function of K2 for a monolayer in the saturation region (n = 0.2,J = 0.2 eV, number of layers. The maximum band occupation for see also Fig. 4). whichferromagnetismexistsdoesnotdependonthefilm thickness,indicatingthat the FMphaseboundaryinthe strongcouplingregimeisthesamein2Dand3D.Alsoin Reducing further the exchange coupling J yields a finite Fig. 7 a comparison between ∆U and T is shown. The C T only for very thin films. C variation of the critical temperature with n is paralleled byacorrespondingvariationof∆U . ∆U variesonly kin pot slowly with n. We find that, apart from a small differ- 4. Anisotropy enceinthemaximumnyieldingferromagnetism,thereis evenquantitativelyagoodagreementbetweenthevalues Inthelastsubsectionwewanttobrieflydiscusstheef- of T and ∆U. C fect ofa single-ionanisotropyonthe ferromagnetictran- InFig. 8weshow∆U and∆U asfunctionsofT kin pot C sition temperature of Kondo lattice films. There exists with the number of layers as implicit parameter. One abundantliteratureonscalingrelationsbetweenthecrit- recognizes a linear dependence between the ferromag- ical temperature and key film parameters as anisotropy netic transition temperature and the energy change. T C strength, exchange coupling, and the film thickness, is again dominantly determined by the difference in ki- mostlyfor Ising-likeorHeisenbergfilms. As we aredeal- netic energy of the FM and of the PM phase. We point out that the almost quantitative agreement of TMRKKY ing with an effective localized spin model, derived from C theKLM,theseresultsshouldalsoapplyinourcase. The and∆U forthemonolayerisduetotheparticularchoice exchange integrals (14) now are, however, T-dependent. oftheanisotropyparameterandthusrathercoincidental. In the following investigation of the functional depen- With an increasing number of layers deviations between dence T (J,K ) we restrict ourselves to a monolayer the two quantities appear. However the magnitudes re- C 2 because in this case the role of anisotropy is most pro- main similar. nounced and the extension to multilayers is straightfor- ward. The effect of the anisotropy strength on the ferromag- 3. TC as a function of film thickness netic transition temperature over the whole range of the intra-atomic exchange coupling J is shown in Fig. 10. Weconcludethissectionbypresentingthecriticaltem- One can distinguish two regimes: As can be seen from peratureasafunctionofthefilmthickness. Asanatural the inset (representing the saturation regime) the well consequence of the results and the discussion just pre- knownlogarithmic dependence of the Curie temperature sented this dependence varies drastically in the different ontheanisotropystrengthK isalsovalidfortheKondo 2 couplingregimesasdemonstratedinFig. 9. Whereasfor lattice model. Already a very small anisotropy is suffi- large J the Curie temperature decreases when reducing cient to obtain a T which is of the order of magnitude C thenumberoflayers,reflectingthefindingsjustdiscussed of Curie temperatures realized for much larger,more re- beforeinFig. 8,wegetacompletelydifferentpicturefor alistic values of K . 2 weak to intermediate strength of J. At J =0.16 the fer- Evaluatingthe formulaforthe Curietemperature(22) romagnetic transition temperature is equal for all films. by solving the corresponding q-integral for the predomi- 9 nantly contributing small wave vectors one gets Curie temperatures were discussed for a variety of pa- rameters like the exchange coupling J, the band occu- 1 J˜ pation n, and for various number of layers. For large ∝ln 1+c T=TC . (27) TC K2 ! enough band occupation we obtain a critical Jc below which ferromagnetism is not stable. In the strong cou- plingregimeT saturatesandincreaseswiththenumber c is a constant ≈ 1 and J˜is the T−dependent effective C of layers. We have not found any significant influence of coupling of one localized spin to all others, i.e. it de- charge transfer on the Curie temperature of KLM films. termines the orderofmagnitudeofthe magnonenergies, which are typically much larger than the anisotropy en- In addition to the MRKKY self-consistent T -results C ergy,J˜>>K2. AnanalogousformulafortheHeisenberg energy calculations of the paramagnetic and ferromag- modelwithconstantexchangecouplingisdiscussed,e.g., netic state were used to discuss both the critical inter- inRef. 40. InourcasetheT-dependenceoftheexchange action strength in the intermediate coupling regime and integrals(14)intheparamagneticregimeisevidentlytoo thecrucialrolethekineticenergyplaysinthestrongcou- weaktocausesignificantdeviationsfromthelogarithmic pling regime. Depending on the values of the exchange dependence, not too surprisingly because this tempera- couplingandthe bandoccupationthe dependence ofthe ture dependence is a mere Fermi softening effect. Curie temperature on the film thickness exhibits quite WhereasinthesaturationregionthescalingofTC with a different behavior. We also discussed the effects of K2 is determined by Eq. (27) as just discussed, in the a single-ion anisotropy on the ferromagnetic transition criticalregionFMbecomesunstableandahigherK2only temperature. Whereas it can significantly elevate TC in slightly reduces Jc. There TC →0 and thus the effective the strong coupling regime it hardly affects the critical couplingJ˜→0,seeEq. (22)wheretheeffectivecoupling interaction strength J . c corresponds to the Jˆ-exchange term. The critical region Inthis workwetooktheratherclassicalspinquantum has a width of ∆J ≈10−1 eV within which the effective coupling typically changes by ∆J˜ ≈ 10−3 eV. Thus a numberS=7/2. This wouldapply forrareearthsystems as,e.g.,Gd. SystemsofcurrentinterestasDMS(S=5/2) ∆K ≈ 10−4 eV should shift J by ≈ 10−2 eV, which is 2 c possessasomewhathigh S,andevenfor the manganites in accordance with the numerical findings in Fig. 10. withS=3/2classicalspinsarebelievedtobeareasonable approximationatleast atlow temperatures.41 Neverthe- less a more profound investigation of the dependence of V. CONCLUSIONS the magnetic properties on the spin quantum number at higher temperatures appears to be highly valuable. WehaveinvestigatedferromagneticKondolatticefilms withafinitebandoccupation. Ourself-consistenttheory Aworthwhileextensionofthepresentworkwouldalso ismadeupbyanelectronicandamagneticpart. Theone be to use other lattice structures and to take additional particleGreenfunctionisobtainedwithinanequationof magnetic configurations like antiferromagnetisminto ac- motion approach. Local moment correlation functions count. A combined bandstructure many-body calcula- are calculated by an effective Heisenberg Hamiltonian tion for Gd could contribute to the much debated issues which results from integrating out the electronic degrees ofanenhancedsurfaceCurie temperatureandelectronic of freedom. surfacestatesinthissubstance. Moreonthemodellevel, We presented results for the temperature dependent making the exchange and anisotropy parameters layer one particle exciation spectrum and discussed charge dependent or analysing the influence of non-magnetic transfer between film layers. 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